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Multiplicity results for \(p\)-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions. (English) Zbl 1179.35134

Summary: The multiple results of positive solutions for the following quasilinear elliptic equation: \(-\Delta_pu= \lambda f(x)|u|^{q-2}u+g(x)|u|^{p^*-2}u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), are established. Here, \(0\in\Omega\) is a bounded smooth domain in \(\mathbb R^N\), \(\Delta_p\) denotes the \(p\)-Laplacian operator, \(1\leq q<p<N\), \(p^*=Np/(N-p)\), \(\lambda\) is a positive real parameter, and \(f,g\) are continuous functions on \(\overline{\Omega}\) which are somewhere positive but which may change sign on \(\Omega\). The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.

MSC:

35J62 Quasilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B09 Positive solutions to PDEs
35J20 Variational methods for second-order elliptic equations
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References:

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